#72286135

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#245225	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

(4,12)(10,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,31)(29,30);(1,2)(3,8,11,6)(4,7)(5,9)(10,14,16,13)(12,15)(17,19)(18,22,24,25)(20,28)(21,27)(23,32,29,31)(26,30);(1,3,5,11)(2,6,9,8)(4,10)(7,13)(12,16)(14,15)(17,20)(18,23)(19,26,27,28)(21,30)(22,31,25,32)(24,29);(1,4,5,12)(2,7)(3,10,11,16)(6,13)(8,14)(9,15)(17,18)(19,25)(20,29,30,23)(21,24)(22,27)(26,31,28,32);(1,3)(2,8)(4,10)(5,11)(6,9)(7,14)(12,16)(13,15)(26,28)(31,32)

Polycyclic presentation:

2^10|b^2=g,c^2=fi,d^2=hi,b^a=bf,c^b=ch,d^a=dhi,d^b=dfi,d^c=dgi,e^b=ehi,e^c=efi,f^b=fj,f^d=fi,g^a=gj,g^c=gi,g^d=gj,g^e=gi,h^b=hi,h^c=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72337579, #72338864, #72338881, …
maximal normal subgroup of	#72337579, #72338864, #72338881, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.6*F.9);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);